Here are the card dimensions:
4.38" Height x 0.75" Thickness x 6.50" Length
The card will fit, your motherboard is Form factor: microATX - 24.4 cm (9.6 inches) x 24.4 cm (9.6 inches) so the video card will not go outside the motherboard area.
One more thing, if you want to buy a video card for games try to save/invest a little more and buy at list something like this: http://www.newegg.com/Product/Product.aspx?Item=N82E168... .
This one is PCI-E 2.0 with will work better with your PCI-E 16x. Also for a PSU a quality 400w or more PSU should be enough.
Even so you're assuming his PSU will be able to sustain 180W . Even though that HP PSU may not be the best, it's more than enough for your system with a 5570. Considering it also lists an E5200 as the CPU, I think that 5570 would be a good match for that system.